Finite Group Databasebeta

Subgroups — #513253

Order 512.

296 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33903	C₂	1	2	yes	maximal
2	#13732	C₂	1	1	yes	maximal, characteristic, complemented
3	#13716	C₂	1	2	yes	maximal, complemented
4	#13715	C₂	1	2	yes	maximal, complemented
5	#13708	C₂	1	2	yes	maximal, complemented
6	#13698	C₂	1	2	yes	maximal, complemented
7	C₄×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#13646	C₂	1	1	yes	maximal, characteristic
9	#13634	C₂	1	2	yes	maximal, complemented
10	#3336	—	2	2	no	—
11	C₂×Q₆₄	C₂²	1	1	yes	characteristic
12	C₂×M₆₄	C₂²	1	2	yes	—
13	#2253	C₂²	1	2	yes	—
14	#2237	—	2	2	no	—
15	#2235	—	2	2	no	—
16	#2229	—	2	2	no	—
17	#2216	C₂²	1	2	yes	—
18	#2216	C₂²	1	2	yes	—
19	C₄×Q₃₂	C₂²	1	2	yes	—
20	D₄×C₁₆	C₂²	1	1	yes	characteristic
21	#2158	C₂²	1	2	yes	—
22	C₂×#366	C₂²	1	2	yes	—
23	#1424	C₂²	1	2	yes	—
24	#1424	C₂²	1	2	yes	—
25	#1423	C₂²	1	1	yes	characteristic
26	#1423	C₂²	1	2	yes	—
27	#1423	C₂²	1	1	yes	characteristic
28	#1423	C₂²	1	1	yes	characteristic
29	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
30	#1416	C₂²	1	2	yes	—
31	#1416	C₂²	1	2	yes	—
32	#1416	C₂²	1	2	yes	—
33	C₂²⋊C₃₂	C₂²	1	2	yes	—
34	C₄×C₃₂	C₂²	1	1	yes	characteristic
35	#552	—	4	2	no	—
36	C₂×Q₃₂	C₂³	1	2	yes	—
37	C₂²×C₁₆	C₂³	1	2	yes	—
38	#499	—	2	2	no	—
39	#485	—	4	2	no	—
40	#483	—	4	2	no	—
41	#479	—	4	2	no	—
42	#467	—	2	2	no	—
43	#467	—	2	2	no	—
44	C₄×Q₁₆	—	2	2	no	—
45	C₈×D₄	D₄	1	1	yes	characteristic
46	#427	—	2	2	no	—
47	C₂×#107	—	2	2	no	—
48	Q₆₄	—	2	2	no	—
49	M₆₄	—	2	4	no	—
50	C₂×C₃₂	C₂³	1	1	yes	characteristic
51	C₂×C₃₂	C₂³	1	2	yes	—
52	C₂×C₃₂	C₂³	1	1	yes	characteristic
53	#367	—	2	2	no	—
54	#367	—	2	2	no	—
55	#366	—	2	2	no	—
56	#366	C₂³	1	2	yes	—
57	#366	C₂³	1	2	yes	—
58	#366	C₂³	1	2	yes	—
59	C₄⋊C₁₆	D₄	1	1	yes	characteristic
60	#358	—	2	2	no	—
61	#358	—	2	2	no	—
62	#358	—	2	2	no	—
63	C₂²⋊C₁₆	D₄	1	2	yes	—
64	C₄×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆	—	2	2	no	—
66	C₂²×C₈	C₂×D₄	1	2	yes	—
67	#126	—	8	2	no	—
68	#125	—	4	2	no	—
69	#125	—	8	2	no	—
70	#123	—	8	2	no	—
71	C₂²⋊Q₈	—	4	2	no	—
72	C₂²⋊Q₈	—	4	2	no	—
73	C₄×Q₈	—	4	2	no	—
74	C₄×D₄	D₈	1	1	yes	characteristic
75	#117	—	4	2	no	—
76	#117	—	8	2	no	—
77	C₂×C₄⋊C₄	—	4	2	no	—
78	Q₃₂	—	2	2	no	—
79	Q₃₂	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
83	C₂×C₁₆	C₂×D₄	1	2	yes	—
84	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
85	#107	—	2	2	no	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#107	—	4	2	no	—
89	#106	—	4	2	no	—
90	#106	—	4	2	no	—
91	C₄⋊C₈	D₈	1	1	yes	characteristic
92	#103	—	4	2	no	—
93	#103	—	4	2	no	—
94	#103	—	4	2	no	—
95	C₂²⋊C₈	D₈	1	2	yes	—
96	C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	C₃₂	—	2	2	no	—
98	C₃₂	—	2	1	no	—
99	C₃₂	#41	1	2	yes	—
100	C₂×Q₈	—	4	2	no	—
101	C₂×D₄	D₁₆	1	1	yes	characteristic
102	C₂²×C₄	—	8	2	no	—
103	C₂²×C₄	C₂×D₈	1	2	yes	—
104	Q₁₆	—	4	2	no	—
105	Q₁₆	—	4	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	C₂×D₈	1	1	yes	characteristic
109	C₂×C₈	C₂×D₈	1	2	yes	—
110	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	8	2	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₄⋊C₄	—	8	2	no	—
119	C₄⋊C₄	—	8	2	no	—
120	C₄⋊C₄	D₁₆	1	1	yes	characteristic
121	C₂²⋊C₄	—	8	2	no	—
122	C₂²⋊C₄	—	8	2	no	—
123	C₂²⋊C₄	—	8	2	no	—
124	C₂²⋊C₄	D₁₆	1	2	yes	—
125	C₄²	—	8	2	no	—
126	C₄²	—	8	2	no	—
127	C₄²	C₂×D₈	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	—	2	1	no	—
130	C₁₆	C₂×#41	1	1	yes	characteristic
131	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
132	C₂³	C₂×D₁₆	1	2	yes	—
133	Q₈	—	8	2	no	—
134	Q₈	—	8	2	no	—
135	D₄	—	4	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	16	4	no	—
143	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
144	C₂×C₄	C₂×D₁₆	1	2	yes	—
145	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
146	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	1	no	—
149	C₈	#547	1	1	yes	characteristic
150	C₈	#578	1	1	yes	characteristic
151	C₂²	—	2	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂²×D₁₆	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	16	2	no	—
159	C₄	—	16	2	no	—
160	C₄	—	16	2	no	—
161	C₄	—	16	2	no	—
162	C₄	#3334	1	1	yes	characteristic
163	C₄	#3417	1	1	yes	characteristic
164	C₂	—	4	2	no	—
165	C₂	#33901	1	1	yes	characteristic
166	C₂	#33981	1	1	yes	characteristic
167	C₂	#33984	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×D₄	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #13732	C₂	semidirect
3. #13716	C₂	semidirect
4. #13715	C₂	semidirect
5. #13708	C₂	semidirect
6. #13698	C₂	semidirect
7. C₄×Q₆₄	C₂	semidirect
9. #13634	C₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#33903	2
#13732	2
#13716	2
#13715	2
#13708	2
#13698	2
C₄×Q₆₄	2
#13646	2
#13634	2